You have two coins, a blue and a red one. You choose one of the coins at random,

Question

You have two coins, a blue and a red one. You choose one of the coins at random, each being chosen with probability
1/2. You then toss the chosen coin twice. The coins are biased: with the blue coin, the probability of heads in any given toss is 0.8, whereas for the red coin it is 0.2.

Let B denote the event that you’ve picked the blue coin.
Let Hi denote the event that the i-th toss resulted in heads for i = {1; 2)

Events H1 and H2 are independent by assumption.

(a) Compute P(H1):
(b) Compute P(H2):
(c) Compute P(H1 B):
(d) Compute P(H1 H2).
(e) Are events H1 and H2 independent? Why?
(f) Compute P(H2 | H1).

Question 2. Coin tossing, again. You have two coins, a blue and a red one. You choose one of the coins at random, each being chosen with prob- ability 1/2. You then toss the chosen coin twice. The coins are biased: with the blue coin, the probability of heads in any given toss is 0.8, whereas for the red coin it is 0.2. » Let B denote the event that you've picked the blue coirn ·Let Hi denote the event that the i-th toss resulted in heads for i = 1,2 Notice that given the choice of a coin, events H and H2 are independent, by assumption. (a) Compute P(H) (b) Compute P(H2) (c) Compute P(Hin B) (d) Compute P(Hi n H2) (e) Are events H and H2 independent? Why? (f) Compute P(H H)

Explanation / Answer

a)P(H1) =P(B)*P(H1|B)+P(R)*P(H1|R) =(1/2)*0.8+(1/2)*0.2 =0.5

b) as above P(H2) =0.5

c) P(H1 nB) =P(B)*P(H1|B) =(1/2)*0.8 =0.4

d) P(H1 H2) =P(B)*P(H1 n H2|B)+P(R)*P(H1 n H2|R) =(1/2)*0.8*0.8+(1/2)*0.2*0.2=0.34

e) as P(H1)*P(H2) =0.5*0.5 =0.25 ; which is not equal to P(H1 n H2) . therefore H1 and H2 are not independent,.

f) P(H2|H1) =P(H1 n H2)/P(H1) =0.34/0.5 =0.68

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